The need to find the inverse of a matrix in matrix algebra is immense. This brings us to one of the easiest ways to find the inverse of a matrix – using elementary row operations.
What are elementary operations of a matrix?
- Interchange of any two rows or two columns. Symbolically written as,
and 
- The multiplication of the elements of any row or column by a non-zero number. Symbolically shown as,
and 
- The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non zero number. Represented as,
and 
Invertible Matrices
A square matrix A is said to invertible if its determinant has a non-zero value, i.e., A is non-singular.
For a square matrix A of order n, its inverse is denoted by 
and has an order n. The product of a matrix and its inverse results to the Identity matrix of order n.
We can observe that, if B is the inverse of A, then A is the inverse of B.
The notion of inverse can be used to calculate an unknown matrix within multiplication.
Example:
To know the values of X, we need to multiply both sides of the equation with the inverse of A. Here, the order of X, A, B is same. 
If A and B are known, then X can be calculated using above steps.
Elementary row operations – Inverse of matrix
Before diving into calculating the inverse of a matrix, let us know more about what operations should be applied to the product AB, if elementary operations (transformations) are performed on X, so that the equation X = AB holds. Row operations carried out on X should also be performed on the first matrix, A, of the product AB. Similarly, if column operations were carried out on X then they should also be carried on the second matrix, B, of the product B.
Procedure
1. We write 
, and then apply a sequence of row operations on A till we get, 
.
Example:
Using elementary operations, calculate the inverse of the matrix
Solution:
We start by writing .
We have been able to write as where B is the inverse of matrix A.
Hence, the inverse is
We can validate the answer by multiplying it with A, as .
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